Homework 7
Math 501
Due October 24, 2014
Exercise 1
Let f : R → R be a function. A point x ∈ R is called a fixed point of
f if f (x) = x. For example 0 and 1 and −1 are fixed points of the function
f (x) = x3 .
Suppose f is differentiable and f 0 (x) 6= 1 for all x ∈ R. Prove that f has at
most one fixed point.
Exercise 2
Let
f1 (x) = |x|,
f2 (x) = x|x|,
f3 (x) = |x|3 .
(a) Prove that f1 is not differentiable at 0 and write a formula for f10 (x) where
it exists.
(b) Find f20 (x) for all x ∈ R and prove that f20 is not differentiable at 0.
(c) Find f30 (x) and f300 (x) for all x ∈ R and prove that f300 is not differentiable
at 0.
Exercise 3
Assume that f : R → R is continuous, and for all x 6= 0, f 0 (x) exists. If
limx→0 f 0 (x) = L exists, does it follow that f 0 (0) exists? Prove or disprove.
Exercise 4
Let f : [a, b] → R be differentiable on (a, b) and suppose that it assumes
a maximum or minimum at some θ ∈ (a, b). Prove that f 0 (θ) = 0. Give a
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counterexample to show that this statement is false if one replaces (a, b) with
[a, b].
Exercise 5
Let f : (a, b) → R be a function. The symmetric derivative of f at x is
equal to the limit
f (x + h) − f (x − h)
lim
h→0
2h
if it exists.
(a) Suppose f 0 (x) exists. Prove that
f (x + h) − f (x − h)
.
h→0
2h
f 0 (x) = lim
[Hint: use the mean value theorem.]
(b) Find a function f and a point x for which f 0 (x) does not exist, but the
symmetric derivative of f at x does exist.
Exercise 6
Let f : R → R be given by
f (x) =
x2 + x x ≥ 0
x2
x < 0.
Prove or disprove: f 00 (x) = 2 for all x ∈ R.
Exercise 7
A function f : (a, b) → R satisfies a Hölder condition of order α if α > 0,
and for some constant H and all u, x ∈ (a, b),
|f (u) − f (x) ≤ H|u − x|α .
The function is said to be α-Hölder, with α-Hölder constant H.
Remark. When α = 1, these functions are sometimes called Lipschitz. It’s
not hard to see that the slopes of all secant lines on the graph of a Lipschitz
√
function can’t be too steep. For example, neither f (x) = x2 nor g(x) = x are
Lipschitz as you can check.
Let f : (a, b) → R be α-Hölder with Hölder constant H.
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(a) Prove that f is uniformly continuous.
(b) Prove that f extends uniquely to a continuous function on [a, b] and prove
that the extension is also α-Hölder with Hölder constant H. [Hint: look
back at a previous homework assignment.]
(c) Prove that if α > 1, then f is constant. [Hint: mean value theorem.]
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